Universal Coefficient Theorem - Corollaries

Corollaries

A special case of the theorem is computing integral cohomology. For a finite CW complex X, is finitely generated, and so we have the following decomposition.

Where are the betti numbers of X. One may check that

, and

This gives the following statement for integral cohomology:

For X a compact, closed, and connected n-manifold, this corollary coupled with Poincaré duality gives that .

Read more about this topic:  Universal Coefficient Theorem